STAT331_Final_Project

I. Introduction

This report investigated the relationship between child mortality and carbon dioxide emissions per capita over the years of 1800 – 2022. Two data sets were acquired from Gapminder, which is an organization that seeks to educate the world’s transmitters of information with up-to-date facts about many metrics of health, infrastructure, sustainability, and many other crucial topics.

The first data set, called child_mortality_data, lists the rates of child death under the age of five per 1,000 births from 197 countries (data begins at 1950 for 12 countries). Each observation corresponds to a single country, and each variable is one year.

The second data set, called co2_pcap_cons, tracks tonnes (metric) of carbon dioxide emissions per capita for 194 countries. There are no NA values in this data set. Each observation corresponds to a single country, and each variable is one year.

The hypothesized relationship between the two variables is that as the amount of CO2 emissions per capita increases for a country, its child mortality rate will drop. While we are not investigating causality, we anticipate that increased CO2 emissions indicate increasing industrialization and modernization within a country, which tends to predict dropping levels of child mortality (Ranganathan et al., 2015).

Code

# Load in packages and .csv files

library(tidyverse)
library(broom)

child_mortality_data <- read.csv(here::here("child_mortality_0_5_year_olds_dying_per_1000_born.csv"))

co2_pcap_cons <- read.csv(here::here("co2_pcap_cons.csv"))

II. Methods

a. Data cleaning

After loading the necessary packages and reading in the two Gapminder files, both data sets were cleaned for ease of analysis. Each data set had its year variables re-designated as characters to standardize the variable type. The X’s in front of each year (carried over from Gapminder) were removed, and each year was then re-designated as a numeric type. Each data set was then pivoted into a long format so that each observation corresponded to a single country for a single year, along with their respective child_mortality and emissions values. These two data sets were named child_mort_long and co2_long.

Finally, these two cleaned data sets were joined into one data set called final_data, which includes both child_mortality and emissions values for 194 countries over the years of 1800 – 2022.

Code

# Convert co2_pcap_cons into long format

co2_long <- co2_pcap_cons %>%
  mutate(across(matches("^X(18|19|20)\\d{2}$"), as.character)) %>% 
  pivot_longer(
    cols = matches("^X(18|19|20)\\d{2}$"), 
    names_to = "year",
    values_to = "emissions"
  ) %>%
  mutate(
    year = str_remove(year, "^X"),
    year = as.numeric(year)
  )


# Convert child_mortality_data into long format

child_mort_long <- child_mortality_data %>%
  mutate(across(matches("^X(18|19|20|21)\\d{2}$"), as.character)) %>% 
  pivot_longer(
    cols = matches("^X(18|19|20|21)\\d{2}$"), 
    names_to = "year",
    values_to = "child_mortality"
  ) %>%
  mutate(
    year = str_remove(year, "^X"),
    year = as.numeric(year)
  )


# Join the data sets

final_data <- child_mort_long %>%
  inner_join(co2_long, by = c("year", "country")) %>%
  mutate(
    child_mortality = as.numeric(as.character(child_mortality)),
    emissions = as.numeric(as.character(emissions))
  )

b. Exploratory analysis

In order to get a sense of how these two variables relate over time, child_mortality and emissions (log transformed) for each country were averaged for each decade starting from the year 1953 through the year 2022. A new variable called Decade was created for this purpose. The resulting mean values are represented in the faceted plot below. Each point represents the mean values over a decade of child_mortality and emissions per capita for a single country.

Code

final_data <- final_data %>%
  mutate(Decade = cut(year, breaks = seq(1953, 2023, 10), 
                      labels = c("1953-1962", "1963-1972",
                                 "1973-1982", "1983-1992",
                                 "1993-2002", "2003-2012",
                                 "2013-2022"),
                      include.lowest = TRUE))


final_data_means <- final_data %>%
  filter(!is.na(Decade)) %>%
  group_by(country, Decade) %>%
  summarize(
    mean_emissions = mean(emissions, na.rm = TRUE),
    mean_child_mortality = mean(child_mortality, na.rm = TRUE),
    .groups = 'drop'
  )


ggplot(final_data_means, aes(x = mean_emissions,
                             y = mean_child_mortality)) +
  geom_point() +
  facet_wrap(Decade ~ ., scales = "free") + 
  labs(title = "Mean Child Mortality vs. Mean Emissions Over Decades (1953 - 2022)",
       x = "Mean Emissions Per Capita (Log Transformed)",
       y = "") +
  scale_x_log10() +
  theme_bw()

Code

# Un-logged
ggplot(final_data_means, aes(x = mean_emissions,
                             y = mean_child_mortality)) +
  geom_point() +
  facet_wrap(Decade ~ ., scales = "free") + 
  labs(title = "Mean Child Mortality vs. Mean Emissions Over Decades (1953 - 2022)",
       x = "Mean Emissions Per Capita",
       y = "") +
  theme_bw()

Each facet in the above plot clearly shows that the global correlation between child_mortality and emissions is negative in every decade since 1953. This consistent pattern supports the hypothesis of an inverse relationship between these variables and provides a solid foundation for further analysis.

III. Analysis

a. Linear regression

To evaluate the relationship between mean_child_mortality and mean_emissions, the final_data set was processed to compute yearly global averages for these variables, resulting in a dataset named yearly_means. Then, a scatterplot was generated, where each point represents the global averages for a given year. An overlaid line of best fit reveals a pronounced negative correlation, indicating that years with higher emissions are generally associated with lower child mortality rates, and vice versa.

Code

# NEED TO DISCUSS - THIS PLOT USES THE WRONG VARIABLES

final_data %>%
  group_by(country) %>%
  summarise(
    mean_child_mortality = mean(child_mortality, na.rm = TRUE),
    mean_emissions = mean(emissions, na.rm = TRUE)
  ) %>%
  ggplot(aes(x = mean_emissions, y = mean_child_mortality)) +
  geom_point() +
  theme_bw() +
  labs(x = "Mean Emissions (Log Transformed)", y = "", 
       title = "Scatter Plot of Mean Child Mortality vs Mean Emissions by Country") +
  geom_smooth(method = "lm", color = "blue") +
  scale_x_log10()

Code

yearly_means <- final_data %>%
  group_by(year) %>%
  summarise(
    mean_child_mortality = mean(child_mortality, na.rm = TRUE),
    mean_emissions = mean(emissions, na.rm = TRUE)
  )


yearly_means %>%
  mutate(period = 25 * floor((year - 1800) / 25) + 1800) %>%
  ggplot(aes(x = mean_emissions, y = mean_child_mortality, color = as.factor(period))) +
  geom_point() +
  theme_bw() +
  labs(x = "Mean Emissions (Log Transformed)", y = "Mean Child Mortality",
       title = "Scatter Plot of Mean Child Mortality vs Mean Emissions by Period",
       color = "Period") +
  geom_smooth(method = "lm", color = "blue") +
  scale_x_log10() +
  scale_color_viridis_d()

Code

# Un-logged
yearly_means %>%
  mutate(period = 25 * floor((year - 1800) / 25) + 1800) %>%
  ggplot(aes(x = mean_emissions, y = mean_child_mortality, color = as.factor(period))) +
  geom_point() +
  theme_bw() +
  labs(x = "Mean Emissions", y = "Mean Child Mortality",
       title = "Scatter Plot of Mean Child Mortality vs Mean Emissions by Period",
       color = "Period") +
  geom_smooth(method = "lm", color = "blue") +
  scale_color_viridis_d()

To extract the linear equation from the above plot, a linear model was created with mean_emissions as the explanatory variable, and mean_child_mortality as the response variable. The resulting table shows the linear analysis.

The regression equation for the model is \(\hat{y}=403.29298−74.17116⋅x\), where \(\hat{y}\) is equal to the estimated mean_child_mortality, and \(x\) is equal to the mean_emissions.

Code

final_data_lm <- lm(mean_child_mortality ~ mean_emissions,
                    data = yearly_means)

tidy(final_data_lm)

# A tibble: 2 × 5
  term           estimate std.error statistic   p.value
  <chr>             <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)       403.       2.88     140.  8.87e-218
2 mean_emissions    -74.2      1.17     -63.4 1.02e-143

---

Test

Code

yearly_means_log <- yearly_means %>%
  mutate(log_mean_emissions = log(mean_emissions))

final_data_lm_log <- lm(mean_child_mortality ~ log_mean_emissions, data = yearly_means_log)

Code

tidy(final_data_lm_log)

# A tibble: 2 × 5
  term               estimate std.error statistic   p.value
  <chr>                 <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)           240.       3.58      67.1 6.43e-149
2 log_mean_emissions    -85.4      2.17     -39.4 5.44e-102

Code

r_yearly_log <- yearly_means_log %>%
  summarize(pearsons_r = cor(mean_child_mortality,
                             log_mean_emissions, method = "pearson")) %>%
  pull(pearsons_r)

Code

final_data_lm_log %>%
  augment() %>%
  ggplot(aes(x = .resid)) +
    geom_histogram(aes(y = ..density..), binwidth = 1, colour = "black", fill = "white") +
    geom_density(alpha = .2, fill = "#FF6666") +
    labs(x = "Residuals", y = "Density",
         title = "Histogram of Residuals with Normal Density Curve") +
    theme_bw()

Code

final_data_lm_log %>%
  augment() %>%
  summarise(
    response_variance = var(mean_child_mortality, na.rm = TRUE),
    fitted_variance = var(.fitted, na.rm = TRUE),
    residuals_variance = var(.resid, na.rm = TRUE)
  ) %>%
  pivot_longer(
    everything(),
    names_to = "Metric",
    values_to = "Variance"
  ) %>%
  mutate(
    Metric = recode(Metric,
      'response_variance' = "Variance of Response",
      'fitted_variance' = "Variance of Fitted Values",
      'residuals_variance' = "Variance of Residuals"
    )
  )

# A tibble: 3 × 2
  Metric                    Variance
  <chr>                        <dbl>
1 Variance of Response        20559.
2 Variance of Fitted Values   18002.
3 Variance of Residuals        2558.

Code

fitted_values_log <- predict(final_data_lm_log)
st_error_log <- sigma(final_data_lm_log)

simulated_data_log <- data.frame(predicted = fitted_values_log) %>%
  mutate(simulated_child_mortality = exp(predicted + rnorm(n(), mean = 0, sd = st_error_log)))

ggplot(simulated_data_log, aes(x = predicted, y = simulated_child_mortality)) +
  geom_point()

End of Test

---

b. Assessing appropriateness of fit

In order to conclude with confidence that appropriateness of using a linear regression for this relationship, the four model conditions of linearity, independence, normality of distribution, and equality of variance must be assessed. Each will be discussed in turn:

Code

r_yearly <- yearly_means %>%
  summarize(pearsons_r = cor(mean_emissions,
                             mean_child_mortality, method = "pearson"))



glance(final_data_lm)

# A tibble: 1 × 12
  r.squared adj.r.squared sigma statistic   p.value    df logLik   AIC   BIC
      <dbl>         <dbl> <dbl>     <dbl>     <dbl> <dbl>  <dbl> <dbl> <dbl>
1     0.948         0.948  32.8     4016. 1.02e-143     1 -1094. 2194. 2204.
# ℹ 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>

1. Linearity:
By looking at the above scatterplot from section III.a, a clear linear pattern is evident. The Pearson’s correlation coefficient (r) is -0.9735707. Given these observations, the first criteria failed to be violated.

2. Independence:
The data were intentionally aggregated to calculate global rates of child mortality and emissions per capita, averaging the values without regard to their country of origin. However, considering that each year’s data likely has a significant causal influence on the data of the following year, this model may be at risk of violating the independence assumption. The temporal nature of the data suggests that observations across years are not entirely independent, which could impact the validity of the independence assumption in our analysis.

3. Normality of residuals:
The histogram below displays the distribution of residuals from the model. The distribution is approximately centered around zero, suggesting that the model does not have systematic bias. The overall shape of the distribution approximates a bell curve, but with a pronounced peak (leptokurtosis) and a moderate right skew. Nonetheless, this does not necessarily violate the assumption of normality, especially considering the large sample size.

Code

final_data_lm |> 
  augment() |> 
ggplot(aes(x = .fitted, y = .resid)) +
  geom_point()

Code

# Histogram

final_data_lm %>%
  augment() %>%
  ggplot(aes(x = .resid)) +
    geom_histogram(aes(y = ..density..), binwidth = 1,
                   colour = "black", fill = "white") +
    geom_density(alpha = .2, fill = "#FF6666") +
    labs(x = "Residuals", y = "",
         title = "Histogram of Residuals with Normal Density Curve") +
    theme_bw()

4. Equal variance of residuals:
The scatterplot below illustrates the distribution of residuals against predicted values, providing insight into the homoscedasticity of the model. It appears that there is an increase in the spread of residuals as the predicted values rise. This pattern suggests a potential violation of the equal variance assumption, which could compromise the reliability of the linear regression analysis.

Code

final_data_lm %>%
  augment() %>%
  summarise(
    response_variance = var(mean_child_mortality, na.rm = TRUE),
    fitted_variance = var(.fitted, na.rm = TRUE),
    residuals_variance = var(.resid, na.rm = TRUE)
  ) %>%
  pivot_longer(
    everything(),
    names_to = "Metric",
    values_to = "Variance"
  ) %>%
  mutate(
    Metric = case_when(
      Metric == "response_variance" ~ "Variance of Response",
      Metric == "fitted_variance" ~ "Variance of Fitted Values",
      Metric == "residuals_variance" ~ "Variance of Residuals",
      TRUE ~ Metric # Fallback, should not be needed
    )
  )

# A tibble: 3 × 2
  Metric                    Variance
  <chr>                        <dbl>
1 Variance of Response        20559.
2 Variance of Fitted Values   19487.
3 Variance of Residuals        1072.

Code

predictions <- predict(final_data_lm)
st_error <- sigma(final_data_lm)

predictions %>%
  data.frame(predicted = .) %>%
  mutate(residuals = rnorm(n(), mean = 0, sd = st_error)) %>%
  ggplot(aes(x = predicted, y = residuals)) +
    geom_point() +
  theme_bw() +
  labs(x = "Predicted values", y = "",
       title = "Residuals vs. Predicted Values")

To be incorporated:

The Adjusted R-squared was 0.9476 for the data indicating that close to 95% of the variability in mean child mortality can be explained by mean CO2 emissions. This indicates that the model is a very good fit for the data. The only issue is the plot of the residuals, since the residuals do not have a random distribution and seem to follow a pattern. This could indicate that the relationship between CO2 emissions and child mortality is non-linear and could lead to issues with other tests on the data.

c. Simulation

Code

noise <- function(x, mean = 0, sd) {
  return(x + rnorm(length(x),
            mean,
            sd)
         )
}

simulations <- map_dfc(.x = 1:1000,
                       .f = ~tibble(simulation = noise(predictions,
                                                       sd = st_error)
                                    )
                       )

Code

colnames(simulations) <- colnames(simulations) |> 
  str_replace(pattern = "\\.\\.\\.",
                  replace = "_")

binded_simulations <- yearly_means |>
  filter(!is.na(mean_child_mortality),
         !is.na(mean_emissions)
         ) |>
  select(mean_child_mortality) |>
  bind_cols(simulations)

head(binded_simulations)

Code

simulated_r_sq <- binded_simulations |>
  map(~ lm(mean_child_mortality ~ .x, data = binded_simulations)
      ) |>
  map(glance) |>
  map_dbl(~ .x$r.squared)

head(simulated_r_sq)

mean_child_mortality         simulation_1         simulation_2 
           1.0000000            0.8907460            0.9015945 
        simulation_3         simulation_4         simulation_5 
           0.8925711            0.8958496            0.8992299 

Code

tibble(simulations = simulated_r_sq) |> 
  ggplot(aes(x = simulations)) + 
  geom_histogram(binwidth = 0.001, color = "black", fill = "skyblue") +
  labs(x = expression("Simulated"~ R^2),
       y = "",
       subtitle = "Number of Simulated Models") +
  theme_bw() +
  scale_y_continuous(breaks = seq(0,
                                  60,
                                  by = 2)
                     )

Conclusion

[text here]

References

Ranganathan et al., 2015 https://www.nature.com/articles/palcomms201533

---

Fixed components

Code

library(knitr)
library(kableExtra)

# Averaging across all years for each country

means_data <- final_data %>%
  group_by(country) %>%
  summarise(
    mean_child_mortality = mean(child_mortality, na.rm = TRUE),
    mean_emissions = mean(emissions, na.rm = TRUE)  
  ) %>%
  mutate(
    log_mean_emissions = log(mean_emissions)
  ) %>%
  ungroup()  



means_data_kable <- kable(means_data, "html", escape = FALSE) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = F, position = "left") %>%
  scroll_box(width = "100%", height = "500px")


means_data_kable

country	mean_child_mortality	mean_emissions	log_mean_emissions
Afghanistan	386.25740	0.0466368	-3.0653660
Albania	273.38637	0.4961345	-0.7009082
Algeria	337.47623	0.7639058	-0.2693108
Andorra	19.21055	2.7854933	1.0244250
Angola	389.17399	0.2111839	-1.5550262
Antigua and Barbuda	198.28354	1.5734978	0.4533010
Argentina	236.32422	1.4339731	0.3604490
Armenia	229.97534	0.7998879	-0.2232837
Australia	164.54354	5.2926009	1.6663098
Austria	225.58197	4.2965874	1.4578211
Azerbaijan	240.90448	1.7380404	0.5527582
Bahamas	182.38238	3.5361121	1.2630278
Bahrain	311.77955	4.4071166	1.4832206
Bangladesh	387.63767	0.0727758	-2.6203720
Barbados	365.35740	0.9733587	-0.0270026
Belarus	223.34018	2.0042735	0.6952817
Belgium	166.45704	8.4054709	2.1288828
Belize	215.76906	0.4507937	-0.7967454
Benin	349.71256	0.1069013	-2.2358489
Bhutan	349.52960	0.1370269	-1.9875780
Bolivia	316.47354	0.3524843	-1.0427492
Bosnia and Herzegovina	259.09130	1.2985919	0.2612805
Botswana	291.26413	0.5983901	-0.5135123
Brazil	297.82691	0.5622511	-0.5758067
Brunei	289.50368	6.3411614	1.8470619
Bulgaria	221.15511	1.8445516	0.6122362
Burkina Faso	374.30762	0.0302556	-3.4980738
Burundi	338.23632	0.0125561	-4.3775524
Cambodia	301.53453	0.1707220	-1.7677189
Cameroon	361.40762	0.0793677	-2.5336636
Canada	180.01933	7.2390493	1.9794899
Cape Verde	292.17982	0.3029641	-1.1941409
Central African Republic	356.56054	0.0342915	-3.3728584
Chad	356.25112	0.0196637	-3.9289821
Chile	299.68090	1.1648655	0.1526056
China	305.78283	0.8371480	-0.1777544
Colombia	274.75874	0.5669058	-0.5675621
Comoros	329.61973	0.0583049	-2.8420686
Congo, Dem. Rep.	341.65291	0.0352735	-3.3446221
Congo, Rep.	306.12960	0.1779821	-1.7260725
Costa Rica	278.65794	0.5615336	-0.5770836
Cote d'Ivoire	346.98834	0.1607130	-1.8281351
Croatia	238.03789	1.3277578	0.2834917
Cuba	239.56677	0.7723139	-0.2583642
Cyprus	209.90538	1.9542960	0.6700300
Czech Republic	220.15224	5.2377982	1.6559012
Denmark	147.32538	4.5645919	1.5183291
Djibouti	342.72825	0.2611031	-1.3428398
Dominica	53.66575	0.3456143	-1.0624317
Dominican Republic	312.21973	0.5868655	-0.5329597
Ecuador	292.73498	0.5138206	-0.6658810
Egypt	317.43543	0.4625695	-0.7709584
El Salvador	326.72377	0.2956413	-1.2186085
Equatorial Guinea	359.86816	0.7619910	-0.2718205
Eritrea	345.57489	0.0729238	-2.6183407
Estonia	205.93753	4.4824439	1.5001684
Eswatini	313.37937	0.2373857	-1.4380692
Ethiopia	357.23049	0.0209955	-3.8634464
Fiji	329.91076	0.4592780	-0.7780995
Finland	186.78229	3.8995740	1.3608673
France	162.37700	4.0272780	1.3930907
Gabon	325.99283	1.2170448	0.1964257
Gambia	372.29821	0.0687848	-2.6767732
Georgia	233.11040	0.8167220	-0.2024565
Germany	229.78448	6.3502646	1.8484965
Ghana	349.55471	0.1608565	-1.8272426
Greece	208.14655	1.8874978	0.6352520
Grenada	206.47848	0.3923049	-0.9357159
Guatemala	363.13229	0.2333453	-1.4552360
Guinea	369.68475	0.0740448	-2.6030844
Guinea-Bissau	348.37444	0.0464484	-3.0694126
Guyana	271.49417	0.7439776	-0.2957444
Haiti	375.11749	0.0527175	-2.9428080
Honduras	288.90807	0.2420179	-1.4187434
Hong Kong, China	21.71986	2.9010045	1.0650571
Hungary	240.81704	2.6974664	0.9923130
Iceland	180.74794	2.8275247	1.0394017
India	351.83229	0.2609552	-1.3434067
Indonesia	327.94081	0.3532466	-1.0405888
Iran	377.38341	1.1978430	0.1805225
Iraq	313.98161	1.1182063	0.1117259
Ireland	152.76081	3.4560807	1.2401352
Israel	253.29700	2.6546502	0.9763129
Italy	243.21148	2.5398744	0.9321146
Jamaica	230.29552	0.8638565	-0.1463486
Japan	219.69170	3.0719865	1.1223244
Jordan	298.61031	0.9714664	-0.0289486
Kazakhstan	317.96915	4.3394350	1.4677442
Kenya	373.66502	0.1628700	-1.8148032
Kiribati	272.47489	0.1575874	-1.8477748
Kuwait	342.07493	9.7526906	2.2775432
Kyrgyz Republic	263.57175	1.1759193	0.1620502
Lao	329.20762	0.1677175	-1.7854743
Latvia	217.02982	1.9157354	0.6501016
Lebanon	299.93682	0.8933408	-0.1127871
Lesotho	309.94036	0.2969283	-1.2142647
Liberia	353.10852	0.1338700	-2.0108864
Libya	302.45426	2.2372018	0.8052259
Liechtenstein	21.24575	2.5136143	0.9217217
Lithuania	251.93484	2.4752242	0.9063310
Luxembourg	202.32336	9.7173722	2.2739152
Madagascar	338.22915	0.0517175	-2.9619593
Malawi	366.96682	0.0703812	-2.6538296
Malaysia	291.41170	1.2340269	0.2102827
Maldives	324.39493	0.3335291	-1.0980250
Mali	412.50090	0.0243901	-3.7135766
Malta	224.87215	2.7239507	1.0020833
Marshall Islands	67.14795	0.5554978	-0.5878907
Mauritania	326.60269	0.1430628	-1.9444717
Mauritius	242.85650	0.8357713	-0.1794003
Mexico	326.68655	1.3734305	0.3173116
Micronesia, Fed. Sts.	215.15202	0.3242287	-1.1263061
Moldova	231.82287	1.2841704	0.2501129
Mongolia	318.70000	1.4798117	0.3919148
Montenegro	246.16987	0.8760897	-0.1322868
Morocco	303.78565	0.3517982	-1.0446975
Mozambique	357.83274	0.1464305	-1.9212044
Myanmar	333.33946	0.1638206	-1.8089832
Namibia	293.19731	0.4525381	-0.7928833
Nauru	69.70000	3.4392242	1.2352459
Nepal	325.18072	0.0465112	-3.0680619
Netherlands	164.92390	4.9606682	1.6015404
New Zealand	144.92924	3.2775067	1.1870830
Nicaragua	357.10045	0.2547892	-1.3673186
Niger	370.58296	0.0228610	-3.7783235
Nigeria	363.45291	0.1572511	-1.8499113
North Korea	332.15157	1.2094664	0.1901792
North Macedonia	258.76874	1.7188565	0.5416592
Norway	122.41036	3.1672063	1.1528499
Oman	322.01018	2.2998296	0.8328350
Pakistan	382.93812	0.1901839	-1.6597640
Palau	37.85342	3.6630762	1.2983033
Palestine	307.80807	0.1181749	-2.1355897
Panama	271.92108	0.4641250	-0.7676014
Papua New Guinea	315.19013	0.1545830	-1.8670244
Paraguay	254.11031	0.2817265	-1.2668187
Peru	282.35157	0.5690000	-0.5638748
Philippines	298.00135	0.3092466	-1.1736161
Poland	231.20291	3.6753722	1.3016544
Portugal	235.22480	1.6005650	0.4703567
Qatar	297.75655	7.3184753	1.9904020
Romania	249.18300	1.7500807	0.5596619
Russia	269.54960	3.4704484	1.2442838
Rwanda	329.05605	0.0239462	-3.7319461
Samoa	218.51166	0.2134529	-1.5443390
Sao Tome and Principe	254.96816	0.1302287	-2.0384631
Saudi Arabia	307.46054	3.9673274	1.3780927
Senegal	395.24439	0.1673049	-1.7879372
Serbia	217.43511	1.5843004	0.4601430
Seychelles	181.79686	0.9933677	-0.0066544
Sierra Leone	429.65291	0.0647040	-2.7379317
Singapore	263.00704	6.4327489	1.8614020
Slovak Republic	225.74915	4.5391211	1.5127334
Slovenia	222.98291	2.5129238	0.9214469
Solomon Islands	357.53587	0.1097085	-2.2099282
Somalia	362.48879	0.0270852	-3.6087678
South Africa	293.02466	2.5586906	0.9394956
South Korea	334.21139	2.1194753	0.7511686
South Sudan	398.13184	0.0370448	-3.2956261
Spain	231.76623	1.9819283	0.6840702
Sri Lanka	272.26969	0.2886413	-1.2425707
St. Kitts and Nevis	59.94247	0.8058386	-0.2158718
St. Lucia	220.13901	0.4149238	-0.8796605
St. Vincent and the Grenadines	230.02691	0.3193408	-1.1414964
Sudan	319.03094	0.1166996	-2.1481526
Suriname	273.07309	1.4609686	0.3790996
Sweden	128.63444	3.7264305	1.3154508
Switzerland	166.26103	4.1204439	1.4159609
Syria	303.85067	0.6301390	-0.4618148
Taiwan	296.09933	2.4094574	0.8794016
Tajikistan	255.55426	0.5052063	-0.6827885
Tanzania	325.12556	0.0730628	-2.6164362
Thailand	302.30291	0.5908296	-0.5262276
Timor-Leste	363.87668	0.0569910	-2.8648614
Togo	336.44709	0.0909552	-2.3973887
Tonga	190.89417	0.2493049	-1.3890785
Trinidad and Tobago	223.89283	4.8734843	1.5838091
Tunisia	338.26951	0.4442825	-0.8112946
Turkey	290.41529	0.9963004	-0.0037064
Turkmenistan	330.32287	4.0312960	1.3940879
Tuvalu	68.19726	0.2638565	-1.3323499
UAE	309.10951	5.5005561	1.7048492
UK	153.93148	8.0481614	2.0854437
USA	177.84022	10.0326951	2.3058493
Uganda	386.88161	0.0422108	-3.1650801
Ukraine	239.93363	2.4806592	0.9085243
Uruguay	242.11081	0.7128924	-0.3384248
Uzbekistan	269.65426	1.9439327	0.6647131
Vanuatu	218.91480	0.2458475	-1.4030437
Venezuela	269.91480	1.9177803	0.6511684
Vietnam	287.26144	0.2582825	-1.3537013
Yemen	422.88161	0.2103677	-1.5588983
Zambia	319.54350	0.5254260	-0.6435459
Zimbabwe	292.04709	0.6607399	-0.4143950

Code

# Over time scatterplot

final_data <- final_data %>%
  mutate(Decade = cut(year, breaks = seq(1953, 2023, 10), 
                      labels = c("1953-1962", "1963-1972",
                                 "1973-1982", "1983-1992",
                                 "1993-2002", "2003-2012",
                                 "2013-2022"),
                      include.lowest = TRUE))


final_data_means <- final_data %>%
  filter(!is.na(Decade)) %>%
  group_by(country, Decade) %>%
  summarize(
    mean_emissions = mean(emissions, na.rm = TRUE),
    mean_child_mortality = mean(child_mortality, na.rm = TRUE),
    .groups = 'drop'
  )


ggplot(final_data_means, aes(x = mean_emissions,
                             y = mean_child_mortality)) +
  geom_point() +
  facet_wrap(Decade ~ ., scales = "free") + 
  labs(title = "Mean Child Mortality vs. Mean Emissions Over Decades (1953 - 2022)",
       x = "Mean Emissions Per Capita (Log Transformed)",
       y = "") +
  scale_x_log10() +
  theme_bw()

Code

# Snapshot Scatterplot

means_data %>%
  ggplot(aes(x = log_mean_emissions, y = mean_child_mortality)) +
  geom_point() +
  theme_bw() +
  labs(x = "Log of Mean Emissions", y = "", 
       title = "Scatter Plot of Mean Child Mortality vs Log of Mean Emissions by Country") +
  geom_smooth(method = "lm", color = "blue")

Code

# New linear model

means_data_lm <- lm(mean_child_mortality ~ log_mean_emissions,
                    data = means_data)

tidy(means_data_lm)

# A tibble: 2 × 5
  term               estimate std.error statistic   p.value
  <chr>                 <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)           260.       4.93      52.7 4.32e-116
2 log_mean_emissions    -31.3      2.98     -10.5 1.07e- 20

Code

# New scatterplot and histogram for residuals

means_data_lm %>%  
  augment() %>%  
ggplot(aes(x = .fitted, y = .resid)) +
  geom_point()

Code

means_data_lm %>%
  augment() %>%
  ggplot(aes(x = .resid)) +
    geom_histogram(aes(y = ..density..), colour = "black", fill = "white") + # Removed binwidth
    geom_density(alpha = .2, fill = "#FF6666") +
    labs(x = "Residuals", y = "Density",
         title = "Histogram of Residuals with Normal Density Curve") +
    theme_bw()

Code

# New variances table

means_data_lm %>%
  augment() %>%
  summarise(
    response_variance = var(mean_child_mortality, na.rm = TRUE),
    fitted_variance = var(.fitted, na.rm = TRUE),
    residuals_variance = var(.resid, na.rm = TRUE)
  ) %>%
  pivot_longer(
    everything(),
    names_to = "Metric",
    values_to = "Variance"
  ) %>%
  mutate(
    Metric = case_when(
      Metric == "response_variance" ~ "Variance of Response",
      Metric == "fitted_variance" ~ "Variance of Fitted Values",
      Metric == "residuals_variance" ~ "Variance of Residuals",
      TRUE ~ Metric # Fallback, should not be needed
    )
  )

# A tibble: 3 × 2
  Metric                    Variance
  <chr>                        <dbl>
1 Variance of Response         6730.
2 Variance of Fitted Values    2458.
3 Variance of Residuals        4272.

Code

predictions_means <- predict(means_data_lm)
st_error_means <- sigma(means_data_lm)

predictions_means %>%
  data.frame(predicted = .) %>%
  mutate(residuals = rnorm(n(), mean = 0, sd = st_error_means)) %>%
  ggplot(aes(x = predicted, y = residuals)) +
    geom_point() +
  theme_bw() +
  labs(x = "Predicted values", y = "",
       title = "Residuals vs. Predicted Values")

Start of new simulation

Code

predictions_means <- predict(means_data_lm)

# Generate simulated residuals
simulated_residuals_means <- rnorm(length(predictions_means), mean = 0, sd = sigma(means_data_lm))

# Create a dataframe with predicted values and simulated residuals
simulated_data_means <- data.frame(predicted = predictions_means, residuals = simulated_residuals_means)

# Plot simulated residuals vs predicted values to assess variance
ggplot(simulated_data_means, aes(x = predicted, y = residuals)) +
  geom_point() +
  theme_bw() +
  labs(x = "Predicted Values", y = "Simulated Response Values",
       title = "Simulated Response Values vs. Predicted Values")

Code

noise <- function(x, mean = 0, sd) {
  return(x + rnorm(length(x), mean, sd))
}

# Generate 1000 simulations
simulations_means <- map_dfc(.x = 1:1000,
                       .f = ~tibble(simulation = noise(predictions_means,
                                                       sd = st_error_means)
                                    )
                       )

Code

colnames(simulations_means) <- str_replace(colnames(simulations_means), pattern = "\\.{3}", replace = "_")

binded_simulations_means <- means_data %>%
  filter(!is.na(mean_child_mortality), !is.na(log_mean_emissions)) %>%
  select(country, mean_child_mortality, log_mean_emissions) %>%
  bind_cols(simulations_means)


head(binded_simulations_means)

# A tibble: 6 × 1,003
  country      mean_child_mortality log_mean_emissions simulation_1 simulation_2
  <chr>                       <dbl>              <dbl>        <dbl>        <dbl>
1 Afghanistan                 386.              -3.07          346.        406. 
2 Albania                     273.              -0.701         341.        295. 
3 Algeria                     337.              -0.269         169.        297. 
4 Andorra                      19.2              1.02          305.        293. 
5 Angola                      389.              -1.56          407.        206. 
6 Antigua and…                198.               0.453         260.         77.7
# ℹ 998 more variables: simulation_3 <dbl>, simulation_4 <dbl>,
#   simulation_5 <dbl>, simulation_6 <dbl>, simulation_7 <dbl>,
#   simulation_8 <dbl>, simulation_9 <dbl>, simulation_10 <dbl>,
#   simulation_11 <dbl>, simulation_12 <dbl>, simulation_13 <dbl>,
#   simulation_14 <dbl>, simulation_15 <dbl>, simulation_16 <dbl>,
#   simulation_17 <dbl>, simulation_18 <dbl>, simulation_19 <dbl>,
#   simulation_20 <dbl>, simulation_21 <dbl>, simulation_22 <dbl>, …

Code

simulation_cols_means <- names(binded_simulations_means)[-(1:3)]

simulated_r_sq_means <- map_dbl(simulation_cols_means, ~ {
  formula <- as.formula(paste("mean_child_mortality ~", .x))
  model <- lm(formula, data = binded_simulations_means)
  summary(model)$r.squared
})

tibble(simulations = simulated_r_sq_means) %>%
  ggplot(aes(x = simulations)) + 
    geom_histogram(binwidth = 0.001, color = "black", fill = "skyblue") +
    labs(x = expression("Simulated"~ R^2),
         y = "Frequency",
         subtitle = "Histogram of Simulated R-squared Values") +
    theme_bw() +
    scale_y_continuous()

Everything in one chunk for GPT

Code

library(knitr)
library(kableExtra)

# Averaging across all years for each country

means_data <- final_data %>%
  group_by(country) %>%
  summarise(
    mean_child_mortality = mean(child_mortality, na.rm = TRUE),
    mean_emissions = mean(emissions, na.rm = TRUE)  
  ) %>%
  mutate(
    log_mean_emissions = log(mean_emissions)
  ) %>%
  ungroup()  


means_data_kable <- kable(means_data, "html", escape = FALSE) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = F, position = "left") %>%
  scroll_box(width = "100%", height = "500px")


means_data_kable

country	mean_child_mortality	mean_emissions	log_mean_emissions
Afghanistan	386.25740	0.0466368	-3.0653660
Albania	273.38637	0.4961345	-0.7009082
Algeria	337.47623	0.7639058	-0.2693108
Andorra	19.21055	2.7854933	1.0244250
Angola	389.17399	0.2111839	-1.5550262
Antigua and Barbuda	198.28354	1.5734978	0.4533010
Argentina	236.32422	1.4339731	0.3604490
Armenia	229.97534	0.7998879	-0.2232837
Australia	164.54354	5.2926009	1.6663098
Austria	225.58197	4.2965874	1.4578211
Azerbaijan	240.90448	1.7380404	0.5527582
Bahamas	182.38238	3.5361121	1.2630278
Bahrain	311.77955	4.4071166	1.4832206
Bangladesh	387.63767	0.0727758	-2.6203720
Barbados	365.35740	0.9733587	-0.0270026
Belarus	223.34018	2.0042735	0.6952817
Belgium	166.45704	8.4054709	2.1288828
Belize	215.76906	0.4507937	-0.7967454
Benin	349.71256	0.1069013	-2.2358489
Bhutan	349.52960	0.1370269	-1.9875780
Bolivia	316.47354	0.3524843	-1.0427492
Bosnia and Herzegovina	259.09130	1.2985919	0.2612805
Botswana	291.26413	0.5983901	-0.5135123
Brazil	297.82691	0.5622511	-0.5758067
Brunei	289.50368	6.3411614	1.8470619
Bulgaria	221.15511	1.8445516	0.6122362
Burkina Faso	374.30762	0.0302556	-3.4980738
Burundi	338.23632	0.0125561	-4.3775524
Cambodia	301.53453	0.1707220	-1.7677189
Cameroon	361.40762	0.0793677	-2.5336636
Canada	180.01933	7.2390493	1.9794899
Cape Verde	292.17982	0.3029641	-1.1941409
Central African Republic	356.56054	0.0342915	-3.3728584
Chad	356.25112	0.0196637	-3.9289821
Chile	299.68090	1.1648655	0.1526056
China	305.78283	0.8371480	-0.1777544
Colombia	274.75874	0.5669058	-0.5675621
Comoros	329.61973	0.0583049	-2.8420686
Congo, Dem. Rep.	341.65291	0.0352735	-3.3446221
Congo, Rep.	306.12960	0.1779821	-1.7260725
Costa Rica	278.65794	0.5615336	-0.5770836
Cote d'Ivoire	346.98834	0.1607130	-1.8281351
Croatia	238.03789	1.3277578	0.2834917
Cuba	239.56677	0.7723139	-0.2583642
Cyprus	209.90538	1.9542960	0.6700300
Czech Republic	220.15224	5.2377982	1.6559012
Denmark	147.32538	4.5645919	1.5183291
Djibouti	342.72825	0.2611031	-1.3428398
Dominica	53.66575	0.3456143	-1.0624317
Dominican Republic	312.21973	0.5868655	-0.5329597
Ecuador	292.73498	0.5138206	-0.6658810
Egypt	317.43543	0.4625695	-0.7709584
El Salvador	326.72377	0.2956413	-1.2186085
Equatorial Guinea	359.86816	0.7619910	-0.2718205
Eritrea	345.57489	0.0729238	-2.6183407
Estonia	205.93753	4.4824439	1.5001684
Eswatini	313.37937	0.2373857	-1.4380692
Ethiopia	357.23049	0.0209955	-3.8634464
Fiji	329.91076	0.4592780	-0.7780995
Finland	186.78229	3.8995740	1.3608673
France	162.37700	4.0272780	1.3930907
Gabon	325.99283	1.2170448	0.1964257
Gambia	372.29821	0.0687848	-2.6767732
Georgia	233.11040	0.8167220	-0.2024565
Germany	229.78448	6.3502646	1.8484965
Ghana	349.55471	0.1608565	-1.8272426
Greece	208.14655	1.8874978	0.6352520
Grenada	206.47848	0.3923049	-0.9357159
Guatemala	363.13229	0.2333453	-1.4552360
Guinea	369.68475	0.0740448	-2.6030844
Guinea-Bissau	348.37444	0.0464484	-3.0694126
Guyana	271.49417	0.7439776	-0.2957444
Haiti	375.11749	0.0527175	-2.9428080
Honduras	288.90807	0.2420179	-1.4187434
Hong Kong, China	21.71986	2.9010045	1.0650571
Hungary	240.81704	2.6974664	0.9923130
Iceland	180.74794	2.8275247	1.0394017
India	351.83229	0.2609552	-1.3434067
Indonesia	327.94081	0.3532466	-1.0405888
Iran	377.38341	1.1978430	0.1805225
Iraq	313.98161	1.1182063	0.1117259
Ireland	152.76081	3.4560807	1.2401352
Israel	253.29700	2.6546502	0.9763129
Italy	243.21148	2.5398744	0.9321146
Jamaica	230.29552	0.8638565	-0.1463486
Japan	219.69170	3.0719865	1.1223244
Jordan	298.61031	0.9714664	-0.0289486
Kazakhstan	317.96915	4.3394350	1.4677442
Kenya	373.66502	0.1628700	-1.8148032
Kiribati	272.47489	0.1575874	-1.8477748
Kuwait	342.07493	9.7526906	2.2775432
Kyrgyz Republic	263.57175	1.1759193	0.1620502
Lao	329.20762	0.1677175	-1.7854743
Latvia	217.02982	1.9157354	0.6501016
Lebanon	299.93682	0.8933408	-0.1127871
Lesotho	309.94036	0.2969283	-1.2142647
Liberia	353.10852	0.1338700	-2.0108864
Libya	302.45426	2.2372018	0.8052259
Liechtenstein	21.24575	2.5136143	0.9217217
Lithuania	251.93484	2.4752242	0.9063310
Luxembourg	202.32336	9.7173722	2.2739152
Madagascar	338.22915	0.0517175	-2.9619593
Malawi	366.96682	0.0703812	-2.6538296
Malaysia	291.41170	1.2340269	0.2102827
Maldives	324.39493	0.3335291	-1.0980250
Mali	412.50090	0.0243901	-3.7135766
Malta	224.87215	2.7239507	1.0020833
Marshall Islands	67.14795	0.5554978	-0.5878907
Mauritania	326.60269	0.1430628	-1.9444717
Mauritius	242.85650	0.8357713	-0.1794003
Mexico	326.68655	1.3734305	0.3173116
Micronesia, Fed. Sts.	215.15202	0.3242287	-1.1263061
Moldova	231.82287	1.2841704	0.2501129
Mongolia	318.70000	1.4798117	0.3919148
Montenegro	246.16987	0.8760897	-0.1322868
Morocco	303.78565	0.3517982	-1.0446975
Mozambique	357.83274	0.1464305	-1.9212044
Myanmar	333.33946	0.1638206	-1.8089832
Namibia	293.19731	0.4525381	-0.7928833
Nauru	69.70000	3.4392242	1.2352459
Nepal	325.18072	0.0465112	-3.0680619
Netherlands	164.92390	4.9606682	1.6015404
New Zealand	144.92924	3.2775067	1.1870830
Nicaragua	357.10045	0.2547892	-1.3673186
Niger	370.58296	0.0228610	-3.7783235
Nigeria	363.45291	0.1572511	-1.8499113
North Korea	332.15157	1.2094664	0.1901792
North Macedonia	258.76874	1.7188565	0.5416592
Norway	122.41036	3.1672063	1.1528499
Oman	322.01018	2.2998296	0.8328350
Pakistan	382.93812	0.1901839	-1.6597640
Palau	37.85342	3.6630762	1.2983033
Palestine	307.80807	0.1181749	-2.1355897
Panama	271.92108	0.4641250	-0.7676014
Papua New Guinea	315.19013	0.1545830	-1.8670244
Paraguay	254.11031	0.2817265	-1.2668187
Peru	282.35157	0.5690000	-0.5638748
Philippines	298.00135	0.3092466	-1.1736161
Poland	231.20291	3.6753722	1.3016544
Portugal	235.22480	1.6005650	0.4703567
Qatar	297.75655	7.3184753	1.9904020
Romania	249.18300	1.7500807	0.5596619
Russia	269.54960	3.4704484	1.2442838
Rwanda	329.05605	0.0239462	-3.7319461
Samoa	218.51166	0.2134529	-1.5443390
Sao Tome and Principe	254.96816	0.1302287	-2.0384631
Saudi Arabia	307.46054	3.9673274	1.3780927
Senegal	395.24439	0.1673049	-1.7879372
Serbia	217.43511	1.5843004	0.4601430
Seychelles	181.79686	0.9933677	-0.0066544
Sierra Leone	429.65291	0.0647040	-2.7379317
Singapore	263.00704	6.4327489	1.8614020
Slovak Republic	225.74915	4.5391211	1.5127334
Slovenia	222.98291	2.5129238	0.9214469
Solomon Islands	357.53587	0.1097085	-2.2099282
Somalia	362.48879	0.0270852	-3.6087678
South Africa	293.02466	2.5586906	0.9394956
South Korea	334.21139	2.1194753	0.7511686
South Sudan	398.13184	0.0370448	-3.2956261
Spain	231.76623	1.9819283	0.6840702
Sri Lanka	272.26969	0.2886413	-1.2425707
St. Kitts and Nevis	59.94247	0.8058386	-0.2158718
St. Lucia	220.13901	0.4149238	-0.8796605
St. Vincent and the Grenadines	230.02691	0.3193408	-1.1414964
Sudan	319.03094	0.1166996	-2.1481526
Suriname	273.07309	1.4609686	0.3790996
Sweden	128.63444	3.7264305	1.3154508
Switzerland	166.26103	4.1204439	1.4159609
Syria	303.85067	0.6301390	-0.4618148
Taiwan	296.09933	2.4094574	0.8794016
Tajikistan	255.55426	0.5052063	-0.6827885
Tanzania	325.12556	0.0730628	-2.6164362
Thailand	302.30291	0.5908296	-0.5262276
Timor-Leste	363.87668	0.0569910	-2.8648614
Togo	336.44709	0.0909552	-2.3973887
Tonga	190.89417	0.2493049	-1.3890785
Trinidad and Tobago	223.89283	4.8734843	1.5838091
Tunisia	338.26951	0.4442825	-0.8112946
Turkey	290.41529	0.9963004	-0.0037064
Turkmenistan	330.32287	4.0312960	1.3940879
Tuvalu	68.19726	0.2638565	-1.3323499
UAE	309.10951	5.5005561	1.7048492
UK	153.93148	8.0481614	2.0854437
USA	177.84022	10.0326951	2.3058493
Uganda	386.88161	0.0422108	-3.1650801
Ukraine	239.93363	2.4806592	0.9085243
Uruguay	242.11081	0.7128924	-0.3384248
Uzbekistan	269.65426	1.9439327	0.6647131
Vanuatu	218.91480	0.2458475	-1.4030437
Venezuela	269.91480	1.9177803	0.6511684
Vietnam	287.26144	0.2582825	-1.3537013
Yemen	422.88161	0.2103677	-1.5588983
Zambia	319.54350	0.5254260	-0.6435459
Zimbabwe	292.04709	0.6607399	-0.4143950

Code

final_data <- final_data %>%
  mutate(Decade = cut(year, breaks = seq(1953, 2023, 10), 
                      labels = c("1953-1962", "1963-1972",
                                 "1973-1982", "1983-1992",
                                 "1993-2002", "2003-2012",
                                 "2013-2022"),
                      include.lowest = TRUE))


final_data_means <- final_data %>%
  filter(!is.na(Decade)) %>%
  group_by(country, Decade) %>%
  summarize(
    mean_emissions = mean(emissions, na.rm = TRUE),
    mean_child_mortality = mean(child_mortality, na.rm = TRUE),
    .groups = 'drop'
  )


ggplot(final_data_means, aes(x = mean_emissions,
                             y = mean_child_mortality)) +
  geom_point() +
  facet_wrap(Decade ~ ., scales = "free") + 
  labs(title = "Mean Child Mortality vs. Mean Emissions Over Decades (1953 - 2022)",
       x = "Mean Emissions Per Capita (Log Transformed)",
       y = "") +
  scale_x_log10() +
  theme_bw()

Code

# Snapshot Scatterplot

means_data %>%
  ggplot(aes(x = log_mean_emissions, y = mean_child_mortality)) +
  geom_point() +
  theme_bw() +
  labs(x = "Log of Mean Emissions", y = "", 
       title = "Scatter Plot of Mean Child Mortality vs Log of Mean Emissions by Country") +
  geom_smooth(method = "lm", color = "blue")

Code

# New linear model

means_data_lm <- lm(mean_child_mortality ~ log_mean_emissions,
                    data = means_data)

tidy(means_data_lm)

# A tibble: 2 × 5
  term               estimate std.error statistic   p.value
  <chr>                 <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)           260.       4.93      52.7 4.32e-116
2 log_mean_emissions    -31.3      2.98     -10.5 1.07e- 20

Code

# New scatterplot and histogram for residuals

means_data_lm %>%  
  augment() %>%  
ggplot(aes(x = .fitted, y = .resid)) +
  geom_point()

Code

means_data_lm %>%
  augment() %>%
  ggplot(aes(x = .resid)) +
    geom_histogram(aes(y = ..density..), colour = "black", fill = "white") + # Removed binwidth
    geom_density(alpha = .2, fill = "#FF6666") +
    labs(x = "Residuals", y = "Density",
         title = "Histogram of Residuals with Normal Density Curve") +
    theme_bw()

Code

# New variances table

means_data_lm %>%
  augment() %>%
  summarise(
    response_variance = var(mean_child_mortality, na.rm = TRUE),
    fitted_variance = var(.fitted, na.rm = TRUE),
    residuals_variance = var(.resid, na.rm = TRUE)
  ) %>%
  pivot_longer(
    everything(),
    names_to = "Metric",
    values_to = "Variance"
  ) %>%
  mutate(
    Metric = case_when(
      Metric == "response_variance" ~ "Variance of Response",
      Metric == "fitted_variance" ~ "Variance of Fitted Values",
      Metric == "residuals_variance" ~ "Variance of Residuals",
      TRUE ~ Metric # Fallback, should not be needed
    )
  )

# A tibble: 3 × 2
  Metric                    Variance
  <chr>                        <dbl>
1 Variance of Response         6730.
2 Variance of Fitted Values    2458.
3 Variance of Residuals        4272.

Code

predictions_means <- predict(means_data_lm)

# Generate simulated residuals
simulated_residuals_means <- rnorm(length(predictions_means), mean = 0, sd = sigma(means_data_lm))

# Create a dataframe with predicted values and simulated residuals
simulated_data_means <- data.frame(predicted = predictions_means, residuals = simulated_residuals_means)

# Plot simulated residuals vs predicted values to assess variance
ggplot(simulated_data_means, aes(x = predicted, y = residuals)) +
  geom_point() +
  theme_bw() +
  labs(x = "Predicted Values", y = "Simulated Response Values",
       title = "Simulated Response Values vs. Predicted Values")

Code

noise <- function(x, mean = 0, sd) {
  return(x + rnorm(length(x), mean, sd))
}

Code

# Generate 1000 simulations
st_error_means <- sigma(means_data_lm)

# Simulate the response variable
simulated_response_means <- map_dfc(.x = 1:1000, .f = ~ {
  tibble(
    simulated_child_mortality = noise(
      x = predict(means_data_lm, means_data),  # Use the original model predictions as a base
      sd = st_error_means                      # Add noise based on the model's standard error
    )
  )
})

# Now, bind the simulated response with the original predictor
binded_simulations_means <- means_data %>%
  select(country, log_mean_emissions) %>%
  bind_cols(simulated_response_means)

# Iterate over each simulation to calculate R-squared values
new_col_names <- c("country", "log_mean_emissions", paste("simulated_child_mortality", 1:1000, sep = "_"))
names(binded_simulations_means) <- new_col_names

# Now we can run the map_dbl to calculate R-squared values
simulated_r_sq_means <- map_dbl(1:1000, ~ {
  # Extract the simulated response for this iteration
  col_name <- paste("simulated_child_mortality", .x, sep = "_")
  # Fit the linear model using the simulated response
  model <- lm(reformulate("log_mean_emissions", response = col_name), data = binded_simulations_means)
  # Extract R-squared value
  summary(model)$r.squared
})

# Plotting the histogram of R-squared values
tibble(simulations = simulated_r_sq_means) %>%
  ggplot(aes(x = simulations)) + 
    geom_histogram(binwidth = 0.001, color = "black", fill = "skyblue") +
    labs(x = expression("Simulated"~ R^2),
         y = "Frequency",
         subtitle = "Histogram of Simulated R-squared Values") +
    theme_bw() +
    scale_y_continuous()